One of Maximum Slice Problems in Codility’s Lessons. [Link]

This is a problem in beginner’s level, and the final answer is really simple. However, I post it as a reminder since I took a long to solve it.

Task Description

A non-empty array A consisting of N integers is given. A pair of integers (P, Q), such that 0 ≤ P ≤ Q < N, is called a slice of array A. The sum of a slice (P, Q) is the total of A[P] + A[P+1] + … + A[Q].

Write a function:

def solution(A)

that, given an array A consisting of N integers, returns the maximum sum of any slice of A.

For example, given array A such that:

A[0] = 3 A[1] = 2 A[2] = -6 A[3] = 4 A[4] = 0

the function should return 5 because:

• (3, 4) is a slice of A that has sum 4,
• (2, 2) is a slice of A that has sum −6,
• (0, 1) is a slice of A that has sum 5,
• no other slice of A has sum greater than (0, 1).

Write an efficient algorithm for the following assumptions:

• N is an integer within the range [1..1,000,000];
• each element of array A is an integer within the range [−1,000,000..1,000,000];
• the result will be an integer within the range [−2,147,483,648..2,147,483,647].

My Answer

• Detected Time Complexity: O(N)

It uses two variables: ‘max’ to represent maximum sum and ‘acc’ for storing cumulative sum. If acc is negative at the current step, it restarts cumulative sum by assigning 0; that is because the next sum would be negatively affected.

def solution(A):
    max = A[0]
    acc = 0
    
    for e in A:
        acc += e
        if acc > max:
            max = acc
            
        if acc < 0:
            acc = 0
                
    return max

Another Answer

We can get the same solution by using 2 max functions. The first max gets cumulative sum for the current slice, and the second max keeps or updates the max slice sum.

def solution(A):
    max_sum = sub_sum = A[0]
    for i in range(1, len(A)):
        sub_sum = max(sub_sum + A[i], A[i])
        max_sum = max(max_sum, sub_sum)
    
    return max_sum